On the plasma membrane, there are Na and K diffuse passively according to their gradients.

Then there is the active Na/K pump that pumps against the respective gradients.

My question is, what would happen if body temp decreases very quickly?

I think that since the Na/K pump is tempreature dependent (Q10=2), that in cold conditions it would stop working. If that stopped working, only the passive diffusion would be at work and since it is temperature independent (Q10=1), the gradients inside and outside the cell will equalize.

Also, since the NA/K pump sets up gradients to transport other molecules like glucose against its gradient, there would be other implications as well.